Linear Sampling

In den nächsten Wochen bis zum 20.2.2020 möchte Anna Hein,
Studentin der Wissenschaftskommunikation am KIT, eine Studie im
Rahmen ihrer Masterarbeit über den Podcast Modellansatz
durchführen. Dazu möchte sie gerne einige Interviews mit Ihnen,
den Hörerinnen und Hörern des Podcast Modellansatz führen, um
herauszufinden, wer den Podcast hört und wie und wofür er genutzt
wird. Die Interviews werden anonymisiert und werden jeweils circa
15 Minuten in Anspruch nehmen. Für die Teilnahme an der Studie
können Sie sich bis zum 20.2.2020 unter der Emailadresse
studie.modellansatz@web.de bei Anna Hein melden. Wir würden uns
sehr freuen, wenn sich viele Interessenten melden würden.


In the coming weeks until February 20, 2020, Anna Hein,
student of science communication at KIT, intends to conduct a
study on the Modellansatz Podcast within her master's thesis. For
this purpose, she would like to conduct some interviews with you,
the listeners of the Modellansatz Podcast, to find out who
listens to the podcast and how and for what purpose it is used.
The interviews will be anonymous and will take about 15 minutes
each. To participate in the study, you can register with Anna
Hein until 20.2.2020 at studie.modellansatz@web.de . We would be
very pleased if many interested parties would contact us.



This is the first of three conversation recorded Conference on
mathematics of wave phenomena 23-27 July 2018 in Karlsruhe.
Gudrun talked to Fioralba Cakoni about the Linear Sampling Method
and Scattering.


The linear sampling method is a method to reconstruct the shape
of an obstacle without a priori knowledge of either the physical
properties or the number of disconnected components of the
scatterer. The principal problem is to detect objects inside an
object without seeing it with our eyes. So we send waves of a
certain frequency range into an object and then measure the
response on the surface of the body. The waves can be absorbed,
reflected and scattered inside the body. From this answer we
would like to detect if there is something like a tumor inside
the body and if yes where. Or to be more precise what is the
shape of the tumor. Since the problem is non-linear and ill posed
this is a difficult question and needs severyl mathematical steps
on the analytical as well as the numerical side.


In 1996 Colton and Kirsch (reference below) proposed a new method
for the obstacle reconstruction problem in inverse scattering
which is today known as the linear sampling method. It is a
method to solve the above stated problem, which scientists call
an inverse scattering problem. The method of linear sampling
combines the answers to lots of frequencies but stays linear. So
the problem in itself is not approximated but the interpretation
of the response is.


The central idea is to invert a bounded operator which is
constructed with the help of the integral over the boundary of
the body.


Fioralba got her Diploma (honor’s program) and her Master's in
Mathematics at the University of Tirana. For her Ph.D. she worked
with George Dassios from the University of Patras but stayed at
the University of Tirana. After that she worked with Wolfgang
Wendland at the University of Stuttgart as Alexander von Humboldt
Research Fellow. During her second year in Stuttgart she got a
position at the University of Delaware in Newark. Since 2015 she
has been Professor at Rutgers University. She works at the Campus
in Piscataway near New Brunswick (New Jersey).


References

F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory
and Transmission Eigenvalues, CBMS-NSF Regional Conference Series
in Applied Mathematics, 88, SIAM Publications, 2016.

F. Cakoni, D. Colton, A Qualitative Approach to Inverse
Scattering Theory, Springer, Applied Mathematical Series, Vol.
188, 2014.

T. Arens: Why linear sampling works, Inverse Problems 20
163-173, 2003. https://doi.org/10.1088/0266-5611/20/1/010

A. Kirsch: Characterization of the shape of a scattering
obstacle using the spectral data of the far field operator,
Inverse Problems 14 1489-512, 1998

D. Colton, A. Kirsch: A simple method for solving inverse
scattering problems in the resonance region, Inverse Problems 12
383-93, 1996.



Podcasts

S. Fliss, G. Thäter: Transparent Boundaries. Conversation in
the Modellansatz Podcast episode 75, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

M. Kray, G. Thäter: Splitting Waves. Conversation in the
Modellansatz Podcast episode 62, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.

F. Sayas, G. Thäter: Acoustic scattering. Conversation in the
Modellansatz Podcast episode 58, Department of Mathematics,
Karlsruhe Institute of Technology (KIT), 2015.